The algebraic $K$-groups of a commutative unital ring can indeed be defined as derived functors, but one needs to work in the context of non-abelian homological algebra in the sense of A. Dold, D. Puppe, Homologie nicht-additiver Funktoren, Ann. Inst. Fourier 11 (1961), and
M. Tierney, W. Vogel: Simplicial resolutions and derived functors, Math. Zeit. 111 (1969).
A useful introduction is given in the book `Non-abelian homological algebra and its applications' by Hvedri Inassaridze (Kluwer, 1997). 

For a group $G$, define $\displaystyle Z_\infty (G)=\lim_{\leftarrow} G/\Gamma_i(G)$, 
where $\{\Gamma_i(G)\}$ is the lower central series of $G$. This defines a functor $Z_\infty: Gr\rightarrow Gr$.
Now Theorem 5.1 in the cited book roughly reads as follows:

Let $L_*Z_\infty$ be the left derived functors of the functor $Z_\infty$. Then $L_i Z_\infty(GL(R))$ is isomorphic to Quillen's $K_i(Q)$.